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Composite numbers n such that a^v + b^v + c^v + ... is prime, where a*b*c* ... is the prime factorization of n and v is the number of primes dividing n (counted with repetition).
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%I #6 Jul 21 2018 19:44:09

%S 6,10,12,14,26,28,34,40,45,52,63,74,75,80,94,96,117,126,134,146,152,

%T 153,165,175,194,206,245,261,268,274,296,320,325,326,333,334,363,376,

%U 384,386,387,399,466,468,475,477,486,490,500,504,507,522,531,536,539,550

%N Composite numbers n such that a^v + b^v + c^v + ... is prime, where a*b*c* ... is the prime factorization of n and v is the number of primes dividing n (counted with repetition).

%C 5166592 generates a 41 digit prime. Some consecutive terms are (74,75) (152,153) (325,326) ... Conjecture: there are infinitely many consecutive values.

%e a(5)=26 because 26=2*13 and 2^2 + 13^2 = 173, a prime.

%t pfnvQ[n_]:=Module[{fi=Flatten[Table[#[[1]],#[[2]]]&/@FactorInteger[n]]}, CompositeQ[n]&&PrimeQ[Total[fi^PrimeOmega[n]]]]; Select[Range[600], pfnvQ] (* _Harvey P. Dale_, Jul 21 2018 *)

%Y Cf. A001222, A082813.

%K easy,nonn

%O 1,1

%A _Jason Earls_, Jun 30 2005